The Borsuk-Sieklucki theorem says that for every uncountable family {X α } α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α = β such that dim(X α ∩ X β ) = n. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is clc n+1 Z , where n ≥ 1, and G is an Abelian group. Let {X α } α∈J be an uncountable family of closed subsets of X. If dim G X = dim G X α = n for all α ∈ J, then dim G (X α ∩ X β ) = n for some α = β.